{"id":5485,"date":"2021-09-14T20:56:21","date_gmt":"2021-09-14T15:26:21","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5485"},"modified":"2021-11-22T19:38:03","modified_gmt":"2021-11-22T14:08:03","slug":"differentiation-of-cosecx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differentiation-of-cosecx\/","title":{"rendered":"Differentiation of cosecx"},"content":{"rendered":"

Here you will learn what is the differentiation of cosecx and its proof by using first principle.<\/p>\n

Let’s begin –<\/p>\n

Differentiation of cosecx<\/h2>\n
\n

The differentiation of cosecx with respect to x is -cosecx.cotx<\/p>\n

i.e. \\(d\\over dx\\) (cosecx) = -cosecx.cotx<\/p>\n<\/blockquote>\n

Proof Using First Principle :<\/h2>\n
\n

Let f(x) = cosec x. Then, f(x + h) = cosec(x + h)<\/p>\n

\\(\\therefore\\)\u00a0 \u00a0\\(d\\over dx\\)(f(x)) = \\(lim_{h\\to 0}\\) \\(f(x + h) – f(x)\\over h\\)<\/p>\n

\\(d\\over dx\\)(f(x)) = \\(lim_{h\\to 0}\\) \\(cosec(x + h) – cosec x\\over h\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(d\\over dx\\)(f(x)) = \\(lim_{h\\to 0}\\) \\({1\\over sin(x + h)} – {1\\over sin x}\\over h\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(d\\over dx\\)(f(x)) = \\(lim_{h\\to 0}\\) \\(sin x – sin(x + h)\\over h sin x sin(x +h)\\)<\/p>\n

By using trigonometry formula,<\/p>\n

[sin C – sin D = \\(2sin ({C – D\\over 2})cos ({C + D\\over 2})\\)]<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(f(x)) = \\(lim_{h\\to 0}\\) \\(2sin ({x – x – h\\over 2})cos({x + x + h\\over 2})\\over h sin x sin (x + h)\\)<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(f(x)) = \\(lim_{h\\to 0}\\) \\(2sin ({-h\\over 2})cos({x + h\/2})\\over h sin x sin (x + h)\\)<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(f(x)) = -\\(lim_{h\\to 0}\\) \\(cos ({x + h\/2})\\over sin x sin(x + h)\\).\\(lim_{h\\to 0}\\) \\(sin(h\/2)\\over (h\/2)\\)<\/p>\n

because, [\\(lim_{h\\to 0}\\)\\(sin(h\/2)\\over (h\/2)\\) = 1]<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(f(x)) = -\\(cos x\\over sin x sin x\\)(1) = -cot x cosec x<\/p>\n

Hence, \\(d\\over dx\\) (cosec x) = =cosecx.cotx<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : What is the differentiation of cosec x + x with respect to x?<\/p>\n

Solution<\/strong> <\/span>: Let y = cosec x + x<\/p>\n

\\(d\\over dx\\)(y) = \\(d\\over dx\\)(cosec x + x)<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(y) = \\(d\\over dx\\)(cosec x) + \\(d\\over dx\\)(x)<\/p>\n

By using cosecx differentiation we get,<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(y) = -cosec x cot x + 1<\/p>\n

Hence, \\(d\\over dx\\)(sec x + x) = -cosec x cot x + 1<\/p>\n

Example<\/strong> <\/span>: What is the differentiation of \\(cosec\\sqrt{x}\\) with respect to x?<\/p>\n

Solution<\/strong><\/span> : Let y = \\(cosec\\sqrt{x}\\)<\/p>\n

\\(d\\over dx\\)(y) = \\(d\\over dx\\)(\\(cosec\\sqrt{x}\\))<\/p>\n

By using chain rule we get,<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(y) = \\(1\\over 2\\sqrt{x}\\)(\\(-cosec \\sqrt{x}.cot\\sqrt{x}\\))<\/p>\n

Hence, \\(d\\over dx\\)(\\(cosec\\sqrt{x}\\)) = -\\(1\\over 2\\sqrt{x}\\)(\\(cosec \\sqrt{x}.cot\\sqrt{x}\\))<\/p>\n


\n

Related Questions<\/h3>\n

What is the Differentiation of cosec inverse x ?<\/a><\/p>\n

What is the Integration of Cosecx ?<\/a><\/p>\n

What is the differentiation of 1\/sinx ?<\/a><\/p>\n\n\n

\n
Next – Differentiation of Exponential Function<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differentiation of secx<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the differentiation of cosecx and its proof by using first principle. Let’s begin – Differentiation of cosecx The differentiation of cosecx with respect to x is -cosecx.cotx i.e. \\(d\\over dx\\) (cosecx) = -cosecx.cotx Proof Using First Principle : Let f(x) = cosec x. Then, f(x + h) = cosec(x …<\/p>\n

Differentiation of cosecx<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[36],"tags":[294,295,275,293],"yoast_head":"\nDifferentiation of cosecx - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn differentiation of cosecx by using first principle.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/differentiation-of-cosecx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentiation of cosecx - 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